A guitar string vibrates in a manner resulting in a standing...

A guitar string vibrates in a manner resulting in a standing wave having a wavelength $\lambda$ being formed in the string. A listener hears the fundamental frequency, $f_{1},$ for this particular string. The string is plucked a second time in a manner that produces the second harmonic for this string. How does the wavelength for the second harmonic compare with the wavelength at the fundamental frequency? (A) $\frac{1}{2} \lambda$ (B) $\frac{\sqrt{2}}{2} \lambda$ (C) $\lambda$ (D) $\sqrt{2} \lambda$ (E) 2$\lambda$

A guitar string vibrates in a manner resulting in a standing wave having a wavelength $\lambda$ being formed in the string. A listener hears the fundamental frequency, $f_{1},$ for this particular string. The string is plucked a second time in a manner that produces the second harmonic for this string. How does the wavelength for the second harmonic compare with the wavelength at the fundamental
frequency?
(A) $\frac{1}{2} \lambda$
(B) $\frac{\sqrt{2}}{2} \lambda$
(C) $\lambda$
(D) $\sqrt{2} \lambda$
(E) 2$\lambda$

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Key Concepts

Relationship between Wavelength and Harmonic Number

In a string fixed at both ends, the length of the string is related to the wavelength by the equation L = n(?/2), where n is the harmonic number. This means that for the fundamental mode (n = 1), the wavelength is twice the length of the string, and for the second harmonic (n = 2), the wavelength is equal to the length of the string. Consequently, the wavelength in the second harmonic is half that of the fundamental mode.

Standing Waves

Standing waves are patterns of vibration that remain in a constant position. They are produced by the interference of two waves traveling in opposite directions. In the context of strings, standing waves form when the wave reflects from fixed ends, causing certain points, called nodes, to remain stationary.

Harmonics

Harmonics are the higher frequency modes of vibration of a system, where the frequencies are integer multiples of the fundamental frequency. For a string fixed at both ends, the fundamental mode is the first harmonic, and subsequent modes (second harmonic, third harmonic, etc.) have additional nodes and antinodes, leading to different wavelengths based on the mode of vibration.

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